# Pre-registered protocol: unified adapter eval (written before any test numbers)

Date: 2026-07-13, after val@499, before step-799 completion. No test-set
number for the unified adapter exists at time of writing.

1. **Primary loop depth: k=2, for both tasks.** Chosen on val CE with the
   tie-breaking rule: prefer the SMALLEST k whose hard-cell val CE is within
   0.01 nats of the best k. (Current val: MBPP hard k2−k4 = 0.004, GSM hard
   k2−k4 = 0.006 → both ties → k=2.) The full k-curve is secondary/descriptive.
2. **Checkpoint selection criterion (scalar, fixed now):** mean of the two
   hard-cell val CEs at k=2, tasks weighted equally:
   crit = (gsm_hard_k2 + mbpp_hard_k2)/2. Lowest crit among saved checkpoints
   wins. (At writing: step 499, crit = (0.475+0.205)/2 = 0.340.)
3. **Primary endpoints:** (a) MBPP test pass@1 hard-bucket at k=2 vs k=0;
   (b) GSM8K test accuracy hard-bucket at k=2 vs k=0. McNemar, paired by item.
   Overall accuracy is secondary (known to be underpowered at n=250/256).
4. **Same-harness rule:** all k, INCLUDING k=0 baselines, measured by the
   prompt-only fast-path scripts (`generate_frozen_prompt`; k=0 = plain
   cached generate inside the same function). No numbers carried over from
   the full-position-loop harness.
5. **Known missing control (not covered by this run):** a same-size,
   no-recurrence adapter (h -> h + MLP(h) at the L13->L14 boundary, no loop,
   no band re-run) trained on identical data/objective. Until it exists,
   "the loop does the work (vs. 1.6M new weights anywhere doing it)" is NOT
   established. Queued as the next training run. Note the k=0 column is
   gated off by construction and is a sanity check only — it is not this
   control.
6. **Symmetric interference check (missing):** dedicated GSM8K prompt-only
   adapter as the reference for "unified costs GSM nothing". Queued. Until
   then the no-interference claim is one-directional (MBPP side only).
7. **Band-location ablation (pre-registered 2026-07-13, before any arm ran).**
   Arms, all else identical (adapter size/init, data, curriculum, k, scripts;
   MBPP): early L2-12, mid-narrow L17-27, late L24-34 (width-matched, 11
   layers); shifted L6-22 (width-matched to the original 17). Reference:
   workspace L14-30 (already run, 3 seeds). Prediction: workspace-centered
   arms (L14-30, L17-27) exceed early/late on hard-bucket pass@1 at k=2-4 by
   a wide margin; shifted intermediate. Falsification: near-parity across
   arms demotes the lens claim from "locates where to loop" to "convenient
   discovery tool"; to be reported either way. Primary readout: hard-bucket
   pass@1 at k=4, e400 checkpoints throughout.
8. **Language commitments for the writeup:** the k0->k1 CE collapse (e.g.
   4.36->0.18) is format/template learning expected from any trained adapter
   and must not be quoted as evidence of routing/planning; informative
   comparisons are within k>=1 cells only. Depth ordering k2 vs k4 deltas
   (0.001-0.006 nats) are inside checkpoint jitter and must be described as
   "k>=2 fits hard items equally well; k=1 slightly worse."

9. **Anchor/entrance sweep (pre-registered 2026-07-14 ~03:00, before any arm
   ran).** Arms: bands (13,30), (12,30), (11,30) — injection point shifted
   up from L14 at fixed tap L30; plus tap-23 = (14,23). All E2B/MBPP, same
   recipe, e400, primary readout hard-bucket pass@1 at k=4. Competing
   predictions: (a) "L14 special" (last full-attention KV-computing layer,
   lens boundary) → anchor-13 drops; (b) "KV-channel count" (anchors 11-13
   add 1-3 extra KV-recomputing attention channels) → holds or improves.
   Body-length confound noted: earlier anchors lengthen the loop body; if
   results shift, run matched-length control (12,28) before interpreting.

10. **L9 discriminator arm (pre-registered 2026-07-14 ~10:00, before running).**
    Band (9,30): anchor at L9 — the only other full-attention, KV-computing
    layer below the boundary — deep in the lens's sensor regime, tap fixed at
    L30. Separates the two cliff explanations: (a) "lens boundary" predicts
    catastrophic (like anchors 11-13: 25-34% overall); (b) "full-attention
    KV-layer entry" predicts partial recovery (clearly above the L11-13
    trend, i.e. >40% overall or hard >25%). Registered prediction: (a) —
    the sensor-region content dominates; layer type does not rescue it.
    Same recipe/checkpoint/eval as the anchor sweep (250 items, ks 0,2,4).

---

# Outcomes vs pre-registrations (scored 2026-07-14, after all arms completed)

1. **k=2 primary depth** — held. All primary comparisons reported at k=2;
   k-curves descriptive. k≥2 plateau confirmed (k=8 gen-eval flat).
2. **Checkpoint criterion** — applied as written for the unified adapter.
   Separately reported: val-CE is a poor proxy for generation accuracy;
   later arms therefore pre-committed to fixed steps (e400) instead.
3. **Primary endpoints (unified adapter, k=2 vs k=0)** — (a) MBPP hard
   3.6% → 28.6% (direction as predicted); (b) GSM hard 0% → 6.3%, overall
   10.5% → 9.0% (no overall win — the math boundary result). Both reported.
4. **Same-harness rule** — held throughout (all final tables fast-path,
   k=0 included).
5. **Missing weights control** — run: trained FF adapter = 17.9% hard,
   exactly the untrained-loop level. Loop-vs-weights gap established.
6. **Symmetric interference check** — run (dedicated GSM adapter);
   mixed-task training regressed both tasks; reported as negative result.
7. **Band-location ablation** — prediction CONFIRMED with a caveat:
   L14-30 hard 43.6% ≫ early L2-12 (23.6%, overall destroyed 22.8%) and
   shifted L6-22 (21.8%, overall 29.0%). Caveat discovered: L17-27 and
   L24-34 are structurally null (KV sharing; k>0 ≡ k=0 bit-identical), so
   the "mid-narrow beats late" half of the prediction was untestable at
   E2B; the 12B replication (no shared KV) carries that weight instead.
8. **Language commitments** — honored in PAPER.md (k0→k1 CE collapse not
   cited as planning evidence; k2-vs-k4 nats described as jitter).
9. **Anchor/entrance sweep** — prediction (a) "L14 special" CONFIRMED:
   anchor-13 hard 17.9%/overall 34.4%; 12: 28.6%/30.8%; 11: 25.0%/25.2%;
   monotone collapse below the boundary. Tap-23 arm died in training
   (never rerun); exits 27/30/32/34 within seed noise, so exit choice is
   free. Matched-length control not needed (results did not shift with
   body length in the informative direction).
10. **L9 discriminator** — registered prediction (a) CONFIRMED: band
    (9,30) overall 14.0-21.4%, hard ≤21.4% — catastrophic, like anchors
    11-13, despite L9 being a full-attention KV-computing layer. The lens
    boundary, not layer type, gates the retrofit.

11. **Recurrent-regime arm (pre-registered 2026-07-15, before training).**
    Huginn-style retrofit on the frozen E2B band: RecurrentAdapter
    (learned A,B init α·I/(1−α)·I + zero-init MLP), h0 = norm-scaled
    noise, log-uniform random depth k∈[1,16], bptt=4, same data/steps/
    checkpoint rule (e400 primary) as all merge arms. Eval ks 0,2,4,8,16,32
    on the 250-item MBPP set. Competing predictions: (a) "amortization is
    intrinsic to frozen-band retrofits" → performance plateaus by k≈4 at
    or below the merge arm's level, no depth-monotone gain; (b) "fixed-
    point behavior was an artifact of our fixed-shallow-k training"
    (Huginn regime transfers) → monotone hard-bucket improvement past k=8
    and reduced noise-seed sensitivity after training. Secondary readout:
    path independence (two noise seeds → output agreement rate) at e400.
    Known risk, stated in advance: 600 steps may be far too little for
    this regime (McLeish et al. use ~50B tokens); a null here bounds the
    cheap-retrofit budget only, not the regime.

12. **Parcae-constrained recurrent arm (pre-registered 2026-07-15, before
    training; Prairie et al. 2026 parameterization).** Same as item 11 but
    A = exp(−Δt·exp(a)) diagonal → ρ(A) < 1 by construction; init exactly
    the α=0.3 merge (verified bit-equal at init). ρ(A) logged every 10
    steps in BOTH arms. Theory-derived predictions, stated in advance:
    (a) contraction ⇒ fixed point is a function of e ⇒ the Parcae arm
    SATURATES in k (no depth-monotone gain) and its converged performance
    is amortizable — if so, our deflationary result is a corollary of
    ρ<1, and our observed k≈3–4 convergence is the geometric rate 0.3^k;
    (b) the UNCONSTRAINED item-11 arm either drifts toward ρ≥1 (watch the
    ρ log: divergent runs should show ρ≥1 before loss spikes) or, if it
    gains monotone depth-performance, does so with ρ near 1 — the edge of
    stability is where genuine iteration must live. Either outcome
    formalizes "the anchor coefficient is the stability dial" as
    "the anchor coefficient is the spectral radius".

13. **Per-depth adapter arm + free-ACT probe (pre-registered 2026-07-15,
    before training).** (a) PerDepthAdapter: one merge adapter per
    iteration (n=4, Bae-style depth-wise relaxation at the entrance;
    breaks time-invariance — LTV, no fixed-point guarantee), standard
    curriculum, e400, eval ks 0,2,4,8. Prediction: lands at or below the
    distill/rung-2 amortization ceiling (~46% hard) because depth-indexed
    weights add content, not state-evolution; exceeding it would show
    per-iteration expressivity was binding and amend the deflationary
    claim. Depths >4 reuse adapter 4 (stated: k=8 cell is then
    fixed-point-like by construction). (b) Free-ACT probe on the standard
    merge arm: record per-item convergence depth (cos>0.9995) at k=8 cap.
    Predictions: accuracy unchanged vs fixed k (post-convergence no-ops);
    mean k_conv ≈ 3; hard-labeled items converge SLOWER than easy ones
    (adaptive compute allocates like ACT without any learned halting
    parameter).

--- Outcome, item 11 (scored 2026-07-15, k=16/32 cells cancelled by
decision after k<=8): PREDICTION (a) SUBSTANTIALLY CONFIRMED, with one
twist. The unconstrained arm left contraction immediately (rho(A):
0.3 -> 3.4 by step 100, plateau ~4.5) yet trained smoothly — per-iteration
norm-matching converts magnitude explosion into directional churn, so
"rho>=1 => divergence" becomes "rho>=1 => divergence OR stationary churn"
under a norm projection. Consequences as predicted: substrate damage
(easy 98.4 -> ~69% at all k>0, far exceeding any contractive arm's tax),
val CE flat k=1..16 (stationary, not progressive), hard bucket at
merge level (35.7/39.3/42.9% at k=2/4/8 — a one-item-per-depth-doubling
crawl that at k=8 reaches what the contractive merge reaches at k=4,
never approaching the amortization ceiling from above). 4x parameters
bought nothing. Depth-monotone computation did not emerge at this budget.

14. **Tied-alpha arm (pre-registered 2026-07-15, before training).**
    TiedAlphaAdapter: x = (1−a)⊙e + a⊙ŝ + MLP([e;ŝ]), a = σ(â) per-dim
    learned, init a=0.3 everywhere (bit-equal to MergeAdapter at step 0,
    verified). B tied to (1−a): convex combination keeps the LTI fixed
    point on the e–ŝ segment (substrate-anchored by construction),
    ρ = max(a) < 1 guaranteed, +d≈1.5K params. Standard curriculum,
    s0 = band(e), e400, eval ks 0,2,4,8 on 250 items. This is the one
    untested cell combining parcae's learnable decay with the merge's
    anchoring. Predictions: (a) substrate fidelity preserved (easy ≈
    merge's 88%, unlike both rec arms' ~70%) because anchoring, not
    ρ, controls fidelity; (b) hard-bucket at merge level (no significant
    gain — per-dim constant α is not where capability lives, per the
    adaptive-α E2B result); (c) learned a drifts slightly DOWN from 0.3
    (as in parcae). If (a) holds while rec arms failed it, the
    fixed-point-location dial is causally isolated: same learnable-decay
    freedom, only the tie to (1−a) differs from parcae.

--- Outcome, item 12 (scored 2026-07-15): prediction (a) CONFIRMED in its
dynamics half, REFUTED in its fidelity half — and the refutation is the
finding. Dynamics: rho stayed in (0,1) throughout (0.300 -> 0.292, the
optimizer drifting MORE contractive when confined to the stable region);
loss trajectory as good as or better than the unconstrained arm at every
checkpoint (the rec arm's flight to rho~4.5 was epiphenomenal — all fit
lives in the MLP); eval saturates completely (hard 42.9/42.9/39.3/39.3/
39.3 at k=2/4/8/16/32, easy flat ~71%). Fidelity: easy items were NOT
preserved (71% vs the merge's 88.5%) despite guaranteed contraction —
substrate fidelity is controlled by fixed-point LOCATION (anchored B +
curriculum), not by rho. Conclusion: stability and fidelity are
independent dials (fig_phase.png); the Parcae constraint delivers exactly
what it promises (robust training, convergence, certified tail gradients)
and exactly nothing more. Item 14 (tied-alpha) is the causal isolation of
the fidelity dial.

15. **Fidelity factorial + capacity control + seed (pre-registered
    2026-07-15 ~03:15, before any of these arms ran; overnight batch).**
    The fidelity loss of both rec arms (easy 88.5 -> ~71%) confounds three
    deltas from the winning merge: (i) learned B, (ii) random-depth
    training instead of the difficulty->depth curriculum, (iii) noise s0.
    Item 14 (tied-alpha) tests (i) with anchoring. New single-variable
    cells, everything else = standard merge recipe (fixed B, band(e) s0,
    curriculum, e400, eval ks 0,2,4,8 on 250 items):
      a. merge+randk  — only (ii) changed (log-uniform k in [1,16], bptt 4).
      b. merge+noises0 — only (iii) changed.
      c. merge h=2048 — capacity control for the per-depth arm (6.4M
         shared vs 6.4M depth-indexed): if per-depth beats the ceiling
         but h2048 does not, time-variation (not capacity) is credited;
         if both do, it was capacity all along.
      d. parcae seed 1 — robustness of the fidelity refutation.
    Predictions: (a) and (b) each cost a few points of easy at most
    (anchored fixed point dominates); neither reproduces the ~17-point
    drop — the culprit is the learned/free B (with item 14 as the
    positive control). h2048 stays at the ceiling (hard <=46%), fidelity
    intact. parcae s1 reproduces easy ~71% within seed noise.

--- Outcome, item 13a (scored 2026-07-15): prediction CONFIRMED — per-depth
lands below/at the ceiling, never above. Detail is instructive: fidelity
preserved throughout (easy 88.5/89.3/86.9 at k=2/4/8 — anchored B), but
hard-bucket content is DEPTH-STRANDED: 17.9% at k=2 (adapters 3-4, which
hold the hard-trained content, never execute), 35.7% at k=4, 42.9% at k=8
— where depths 5-8 reuse adapter 4, i.e. the architecture reverts to
shared-map iteration and the fixed-point mechanism collects the remaining
gain. Time-variation adds a fragility (content unavailable except at its
training depth) and no capability; map-sharing is load-bearing for the
anytime-usable gain. Depth-4 adapter overfit visible in val (hard k4 CE
0.188@99 -> 0.371@599) — LTV concentrates small-pool overfitting into
single depths.

--- Outcome, item 13b (scored 2026-07-15): accuracy prediction CONFIRMED
(k=8 halt run 52.0/90.2/42.9 = plateau level); convergence predictions
REFUTED. Per-item state-cosine (thresh 0.9995, k=8 cap): k_conv
distribution 4:3, 5:57, 6:47, 7:17, never-within-8:126 — mean ~7, and NO
difficulty gradient (easy 7.01 vs hard 7.00). The earlier "bit-exact by
k~3-4" was the single dynamics-probe example, not the population: outputs
plateau by k~2-4 while the state keeps drifting at 1e-3..1e-4 cosine
scale; the fixed point is an OUTPUT-stable orbit (suffix layers + decode
wash out residual state motion), not a literal state fixed point for most
prompts. Free-ACT via state-cosine therefore yields no early exit at this
threshold, and no ACT-like difficulty allocation falls out for free —
output-level halting signals would be needed. Paper's dynamics claims
softened accordingly.

--- Outcome, item 14 (scored 2026-07-15): ALL THREE PREDICTIONS CONFIRMED.
(a) Fidelity fully preserved: easy 93.4/91.0/90.2 at k=2/4/8 (merge:
92.6/88.5; parcae with identical decay freedom but untied B: ~71%) —
the free B is causally isolated as the fidelity culprit, the anchoring
tie as the protection. (b) Hard at merge level exactly (35.7/42.9/39.3 =
merge's k-curve within noise); no gain from the freedom. (c) Learned a
essentially unmoved: mean 0.298, range [0.285, 0.310], 0/1536 dims moved
>0.05 from init — the anchor coefficient is not a useful learnable DOF;
hand-tuned 0.3 was already optimal. Recipe consequence: fixed-alpha
anchored merge is the recommended design; learnable-alpha safe but
pointless, learnable-B harmful, per-depth strands the gain.

16. **Code→GSM8K cross-task transfer (pre-registered 2026-07-15 ~14:10,
    before running).** The MBPP-trained loop adapter (adapter_code, s0) and
    the noise-s0 variant evaluated on GSM8K test (n=256, prompt-only loop,
    same harness as eval_gsmonly). Extends the transfer-distance ladder
    (HumanEval tie -> LCB trained-hurts) across tasks. Predictions:
    (a) hard-bucket gain ~0 (plan content is task-local; GSM8K needs
    evolving state, not static plans); (b) easy items damaged at k>0
    (~93 -> 50-70%), comparable to or worse than the GSM-trained merge —
    substrate damage on GSM8K is perturbation-driven and content-agnostic;
    (c) overall at k>0 below k=0 (no rescue). If instead hard gains
    appear (>5 points), plan-shaped content is partially task-general —
    would weaken the task-local claim from LCB.

--- Amendment to item 15 (2026-07-15 ~13:15): noise-s0 arm EXCEEDED
prediction (b) upward: hard 50.0/53.6/50.0 at k=2/4/8 with easy 88-90%
— nominally the best hard cells of the project (merge best 46.4; seed
mean 37.5±5.5). Paired vs tied-alpha (only same-day per-item baseline):
discordants 5-1/3-0/3-0 in noise-s0's favor, each k p≈0.22-0.25 at n=28
— consistent direction, not individually significant. Denoising
interpretation: training the loop to reach the fixed point from noise
regularizes the content. SEED ARMS QUEUED (s1, s2, same recipe/eval,
pre-registered here): if seed-mean hard(k=4) > 46.4 (the merge's best
single cell), the recommended recipe gains noise-s0; if seed mean falls
back into 37-46, it was a lucky seed.

--- Outcome, item 15c (h2048 capacity control, scored 2026-07-15): the
per-depth exoneration is CLEAN — shared 6.4M params reach hard 42.9/53.6/
50.0 at k=2/4/8 vs per-depth's 17.9/35.7/42.9 at the same capacity;
time-variation is strictly worse than weight-sharing at matched params.
Fidelity prediction confirmed and exceeded (easy 95.1% at k=2 — best
looped fidelity of the project; 90.2% at k=4/8). Ceiling prediction
(hard <= 46%) REFUTED UPWARD like noise-s0: k=4/8 at 53.6/50.0. Two
independent variations (noise s0, 4x MLP) now sit at 50-54% where the
original merge reached 46.4 — suggests 46.4 was an UNDER-estimate of the
recipe family's level, not a ceiling it defined. The distill-parity
deflation claim is unaffected statistically (53.6 vs 45.7 at hard n=28
is within noise) but the language "every regime tops out at the same
ceiling" should become "at the same level within noise" — pending the
noise-s0 seed arms.

--- Outcome, item 15d (parcae seed 1, scored 2026-07-15): CONFIRMED —
the fidelity refutation replicates. easy 70.5/73.0/72.1 at k=2/4/8
(seed 0: 72.1/71.3/70.5); hard 32.1/39.3/35.7 (seed 0: 42.9/42.9/39.3,
ordinary seed spread at n=28). Two-seed conclusion: contraction-with-
free-B loses ~17 points of easy items regardless of seed; the phase
diagram's Parcae point is solid.

--- Outcome, item 16 (code->GSM8K transfer, scored 2026-07-15): ALL THREE
PREDICTIONS CONFIRMED, emphatically. MBPP-trained loop on GSM8K: hard
0.8-1.6% at every k (prediction a: ~0 gain — plan content is task-local);
easy 93.1 -> 27.6-44.8% (prediction b: damaged, in fact WORSE than the
GSM-trained merge's 48%); overall strictly below k=0 at every k>0
(prediction c). noise-s0 variant identical (easy 34.5, hard 1.6). The
transfer-distance ladder ends cleanly: near (HumanEval) tie, far-code
(LCB) trained-hurts, cross-task (GSM8K) trained-content actively toxic
while gaining nothing. Task-locality of the learned content is now a
three-point monotone result.

--- Closure of the item-15b/15c "ceiling nudged upward" question
(2026-07-15, after ns seeds): LUCKY SEED, per the pre-registered rule.
noise-s0 hard(k=4) across seeds: 53.6 / 39.3 / 35.7 -> seed mean 42.9,
inside the 37-46 band. Fidelity across seeds intact (easy 90.2-94.3 —
the factorial conclusion is seed-robust); the 50-54% cells (ns seed 0,
h2048 single seed) were upper-tail draws of the same distribution the
merge's 46.4 came from. No recipe amendment; the abstract's original
"same level within noise" framing stands; single-cell records are not
levels — only seed means are.

17. **GSM-only, current recipe (pre-registered 2026-07-15 ~20:45, before
    running).** train_merge_unified.py --tasks gsm: MergeAdapter, prompt-
    only loop, curriculum, GSM8K data ONLY — removes the mixed-task
    interference confound from the adapter_uni run, completing the
    "winning recipe trained on GSM" question. Eval: prompt-only, n=256,
    ks 0,1,2,4, e400. Predictions: (a) hard <= 10% at every k (supervision
    density is structural: ~3 answer tokens; the recipe's dense-output
    ingredient cannot exist here); (b) easy damaged at k>0 (to 40-70%);
    (c) overall never beats k=0. If hard exceeds 15% or overall beats
    k=0, task interference in the mixed run was masking a real GSM
    capability — would reopen the GSM chapter.

    Scope note (item 17): the design-space arms of items 11-15 are NOT
    crossed with GSM8K, deliberately. Exclusion by dominance: fidelity-
    failing regimes (rec, parcae) cannot improve on a task MORE fidelity-
    fragile than MBPP; architecture-failing (per-depth) and equivalent
    (tied-alpha -> merge) and k-placement-only (randk) and same-family
    (noise-s0, h2048) variants have no mechanism by which task change
    could invert their MBPP verdict. Only the recipe family's best member
    (this item) is informative on GSM8K.

--- Outcome, item 17 (GSM-only, current recipe, scored 2026-07-15):
predictions (a) and (b) CONFIRMED, (c) nominally exceeded but not
meaningfully. hard 8.7/5.5/4.7% at k=1/2/4 (below the 10% bar; nowhere
near the 15% reopen threshold); easy 93.1 -> 48-52% at k>0; overall
11.7/10.9/10.2 vs k0's 10.5 — the k=1 cell is +1.2 points nominal
(~3 items at n=256, not significant), the rest below. Removing the
mixed-task interference bought ~2 points over adapter_uni (9.4 -> 11.7
at k=1) — interference was real but marginal, not masking a capability.
The GSM8K chapter is closed: the recipe family's best member, trained
on GSM alone in the correct regime, delivers no usable gain and the
standard fidelity damage; combined with the scope note, the boundary
claim (structural: supervision density + state-evolution bottleneck)
is fully supported.

18. **E1: learned per-prompt halting gate (pre-registered 2026-07-16
    ~00:20, before any arm runs; PLAN_SELFPACED.md).** HaltingMergeAdapter:
    frozen-recipe merge + ACT-style halting head on the last prompt
    position's workspace state; soft state-mixture training, CE + lambda *
    E[iters], penalty warmup at step 100; NO difficulty curriculum (mixed
    batches — the gate must discover the allocation). k_max=4, e400/e600
    checkpoints, deploy = sequential halting at 0.5 cumulative mass,
    generation via frozen-prompt at per-item k*. Arms: lambda in
    {0, 1e-3, 1e-2}, seed 0. Eval: 250 items, vs anchors k=0 (0.488),
    uniform merge k=4 (0.512/0.885/0.464), probe-gate E0 (0.520/0.975/0.286).
    Predictions: (a) some lambda gives overall >= 0.512 at mean E[k] <=
    2.4 (60% of uniform-4); (b) easy >= 0.95 at that lambda; (c) k*-vs-hard
    point-biserial r > 0.3; (d) hard >= 0.286 (beats E0's frozen probe).
    Collapse (E[k] pinned at 1 or 4 for all lambda) falsifies E1 and
    triggers the plan's kill criterion. lambda=0 control isolates whether
    the CE gradient alone moves the gate (expected: barely — penalty
    provides the pressure).

    Item 18 amendment (2026-07-16 ~23:45, before results): arms run on a
    rented 4xH100 node in parallel instead of the Spark queue; a fourth
    arm (lambda=1e-3, seed 1) is added for immediate seed replication of
    the expected-winner penalty. Spark's queued gate jobs will be dropped
    to avoid duplication. Everything else per registration.

--- Outcome, item 18 (scored 2026-07-16 ~00:40): predictions (b), (c)
REFUTED, (a) marginal miss, (d) trivial pass. All arms converge to
UNIFORM depth (lambda 0/1e-3/1e-2 -> E[k] 4/2-or-4/1; the two 1e-3 seeds
picked different plateaus — degenerate penalty landscape), r = 0.000
everywhere. Mechanism identified and consistent with prior findings:
teacher-forced CE is depth-flat (stationarity), so CE provides no
per-item depth gradient; the penalty alone cannot teach selectivity.
The state DOES carry the signal (E0 probe: train acc 1.0) — the failure
is the training signal, not the representation. E1-as-designed is dead;
kill criterion NOT fully triggered (E2 untested, and the mechanism
points at a repair).

19. **E1b: label-supervised halting head (pre-registered 2026-07-16
    ~00:45, before running).** Freeze the curriculum merge (adapter_code
    s0); train ONLY the halting head (BCE): target halt=0 at iterations
    below the label's depth (easy->1, hard->4, per STaR label), halt=1 at
    or above it. 300 steps, mixed batches, head-only params. Eval: gated
    eval as item 18, n=250. Predictions: (a) r(k*, hard) > 0.5 (the head
    is a trained difficulty classifier now); (b) easy >= 95% at k*=1
    (near-E0's 97.5); (c) hard >= 35.7% (>= best uniform arm, via better
    recall than E0's frozen probe: more than 18/28 hard items routed
    deep); (d) overall >= 52.0 at E[k] <= 2.2. If (c) fails while (a,b)
    hold, halting-head recall saturates at probe level and gate quality,
    not gate training, is the binding constraint.

--- Outcome, item 19 / E1b (scored 2026-07-16 ~01:15): prediction (c)
CONFIRMED (hard 39.3 >= 35.7 at mean k* 2.18), (a) FAILED at r=0.217
(selectivity real — hard routed 2x deeper than easy (2.18 vs 1.08), the
program's first nonzero gate correlation — but weak at deploy), (b,d)
FAILED for a traced design reason: halted_k_per_item lacked k*=0, so easy
items were forced through >=1 iteration and landed on the merge's WORST
easy depth (k=1: 85.2%); E0's 97.5% came precisely from k=0 routing.
E1c amendment (pre-registered before running, same session): pre-loop
halt consult on s_0 enabling k*=0; targets easy->0, hard->4; threshold
0.5 unchanged (calibration deferred unless E1c misses). Predictions:
easy >= 95%, hard >= 35.7%, r >= 0.4, overall >= 51.2 at E[k] <= 1.5.

--- Outcome, item 19 / E1c (scored 2026-07-16 ~01:50, Spark re-run):
prediction (b) CONFIRMED — easy 95.9% with mean k*=0.11 (the k=0 routing
fix worked perfectly for easy items); (a) FAILED (r=0.220, unchanged);
(c) FAILED HARD — hard 21.4% at mean k*=1.5: with the pre-loop halt
consult, weakly-discriminated hard items now exit at k*=0/1 where before
they got >=1 iteration; the recall problem became more expensive, not
better. (d) FAILED (overall 50.0 at E[k]=0.74). Net Pareto: E1c =
(50.0 overall, 95.9 easy, 21.4 hard, 0.74 mean iters — 82% compute
saved); E0 probe-gate = (52.0, 97.5, 28.6, ~2.2) still dominates on
accuracy. The learned-head line has ONE identified un-tried knob:
deploy-threshold calibration on val for hard-recall (the 0.5 threshold
is arbitrary; raising it routes more items deep, trading easy tax for
hard recall — a tunable curve E0 cannot offer). E1 arc summary for
PLAN_SELFPACED: gating machinery works end-to-end, easy-item protection
and compute savings are demonstrated and cheap; difficulty-selective
DEPTH allocation remains unsolved at 3K-param-head scale — binding
constraint is classifier quality on the k=0/s0 state, exactly where E0
started. Next per plan: threshold sweep (cheap) before any E2.

20. **E1 threshold curve + oracle bound (pre-registered 2026-07-16 ~02:15,
    before running).** Phase 1: record E1c head's halt probabilities per
    test item (one GPU pass). Phase 2: per-item outcomes for the frozen
    curriculum merge at k=0/1/2/4 (four generation sweeps, tag merge_lut —
    doubles as the reusable gate-evaluation lookup table and supplies the
    long-missing per-item logs for the canonical merge). Phase 3 (offline):
    gated accuracy at thresholds .3-.99 by composing k*(theta) with the
    lookup; plus the ORACLE gate (best k per item) = the ceiling any gate
    can reach with this merge. Predictions: (a) some theta gives hard >=
    32% with easy >= 93% and E[k] <= 2.2 (dominating E0 on compute at
    comparable accuracy); (b) the curve is monotone in theta; (c) oracle
    overall >= 55% — if so, gate-quality headroom is large and further
    gate work is justified; if oracle < 53%, gating this merge is nearly
    saturated and the program pivots to E2 or closes.

--- Outcome, item 20 (scored 2026-07-16 ~02:35): (b) CONFIRMED — clean
monotone threshold curve (hard 7->50%, easy 96.7->87.7%, E[k] 0.43->2.63
across theta .3->.99). (a) FAILED — no theta reaches easy>=93 AND
hard>=32; at matched easy the E0 frozen probe dominates the entire
learned-head curve: the BCE-trained 3K head is strictly worse than the
class-balanced logistic probe it was meant to replace. (c) CONFIRMED,
emphatically: ORACLE gate = 59.6 overall / easy 100% / hard 64.3% at
E[k]=0.24. Key insight: hard items are DEPTH-DIVERSE — 18/28 solvable at
some k in {0,1,2,4} but no single k solves more than 13; a third of the
hard bucket lives in per-item depth selection. Program continues per
rule; binding constraint quantified: gate quality is worth ~9.6 overall
points (50.0 deployed vs 59.6 oracle). Also noted: the LUT re-run of the
canonical merge shows small systematic drift vs the Jul-13 eval (k4 hard
46.4 identical, k1/k2 hard 3 items lower) — the LUT (per-item, single
harness run) is now the canonical reference. Next candidates, in cost
order: (i) deploy E0's probe AS the gate against the LUT (free,
offline); (ii) stronger classifier (multi-position features, more data,
calibrated threshold); (iii) oracle-gap error analysis on the hard items
no fixed k solves but some k does.

21. **E2 stage A: dense short-CoT supervision through the carry
    whiteboard, GSM8K (pre-registered 2026-07-16 ~02:55, before running;
    PLAN_SELFPACED E2 / the hybrid from the internalization discussion).**
    Prep: harvest TERSE verified CoTs ("at most 3 short steps", answer-
    verified, STaR filter) for GSM train. Arms: (A) carry regime
    (k=2 prefill, pauses easy p=2 / hard p=6) trained with CE on
    scratchpad+answer (~30-60 dense tokens — the ingredient every latent
    GSM arm lacked); (B) CONTROL: identical supervision, feedforward
    adapter, no recurrence. Eval: GSM test 256, grid 0:0 (base), 2:2,
    2:6; e400 checkpoints. Predictions: (a) arm A beats every previous
    GSM arm's overall (>12.1%) — dense supervision is the binding fix;
    (b) the A-vs-B delta isolates the whiteboard: if A > B by >=3 points
    overall, recurrence adds value beyond visible-scratchpad training;
    if A ~= B, the scratchpad text alone carries it (deflation, GSM
    edition); (c) easy-bucket damage smaller than answer-only carry's
    (83->45%) because training and deployment output formats now match.
    Honest note: arm outputs are VISIBLE tokens (~40) — this is the
    budget-CoT-with-loop hybrid, a scope change from latent planning,
    run at Nils's explicit direction ("do gsm8k and such").
